Chapter 5
Finding Antiderivatives and
Evaluating Integrals
5.1 Constructing Accurate Graphs of Antiderivatives
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• Given the graph of a function’s derivative, how can we construct a completely
accurate graph of the original function?
• How many antiderivatives does a given function have? What do those antiderivatives all have in common?
• Given a function f , how does the rule A(x) =
x
0
f (t) dt define a new function A?
Introduction
A recurring theme in our discussion of differential calculus has been the question “Given
information about the derivative of an unknown function f , how much information can we
obtain about f itself?” For instance, in Activity 1.22, we explored the situation where the
graph of y = f ′ (x) was known (along with the value of f at a single point) and endeavored
to sketch a possible graph of f near the known point. In Example 3.1 – and indeed
throughout Section 3.1 – we investigated how the first derivative test enables us to use
information regarding f ′ to determine where the original function f is increasing and
decreasing, as well as where f has relative extreme values. Further, if we know a formula
or graph of f ′ , by computing f ′′ we can find where the original function f is concave
265
Finding Antiderivatives and
Evaluating Integrals
5.1 Constructing Accurate Graphs of Antiderivatives
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• Given the graph of a function’s derivative, how can we construct a completely
accurate graph of the original function?
• How many antiderivatives does a given function have? What do those antiderivatives all have in common?
• Given a function f , how does the rule A(x) =
x
0
f (t) dt define a new function A?
Introduction
A recurring theme in our discussion of differential calculus has been the question “Given
information about the derivative of an unknown function f , how much information can we
obtain about f itself?” For instance, in Activity 1.22, we explored the situation where the
graph of y = f ′ (x) was known (along with the value of f at a single point) and endeavored
to sketch a possible graph of f near the known point. In Example 3.1 – and indeed
throughout Section 3.1 – we investigated how the first derivative test enables us to use
information regarding f ′ to determine where the original function f is increasing and
decreasing, as well as where f has relative extreme values. Further, if we know a formula
or graph of f ′ , by computing f ′′ we can find where the original function f is concave
265